Jussi Jokinen and Patrick Dwyer also scored for Carolina, which went 4-1-1 to conclude a six-game road trip.
With Carolina on the power play, Tlusty flipped a shot toward the net that bounced off Devils defenseman Bryce Salvador and over goalie Martin Brodeur at 7:25. Eric Stall picked up an assist, extending his point-scoring streak to 10 games.
Tlusty later added an empty-netter to seal the win.
SENATORS 2, SABRES 0
In Ottawa, Craig Anderson made 42 saves and Erik Karlsson and Erik Condra scored goals less than two minutes apart in the second period as the Senators defeated Buffalo.
Anderson picked up his second shutout of the season as the Senators snapped a mini two-game slide.
Ryan Miller had 29 saves in the loss.
LIGHTNING 4, CANADIENS 3 (SO)
In Tampa, Fla., Carey Price stopped all three shots in the shootout, David Desharnais had the lone goal in the tiebreaker, and Montreal rebounded after blowing a three-goal lead.
Victor Hedman scored twice and Sami Salo added a power-play goal during the final six minutes of regulation as the Lightning tied it at 3.
RANGERS 4, BRUINS 3 (SO)
In Boston, Ryan Callahan scored as the final participant in the shootout and New York recovered after blowing a three-goal, third-period lead.
Henrik Lundqvist stopped 37 shots, plus three of the four attempts in the shootout. Carl Hagelin scored for the second game in a row, and Derek Stepan and Anton Stralman each added an unassisted goal for the Rangers.
CAPITALS 6, PANTHERS 5 (OT)
In Sunrise, Fla., Troy Brouwer scored 32 seconds into OT for Washington, who rallied from a late two-goal deficit.
Washington trailed 5-3 before Eric Fehr scored on a redirect at 14:53 of the third period. Alex Ovechkin then scored a power-play goal with 2:49 left to tie it.
PREDATORS 1, SHARKS 0 (OT)
In Nashville, Tenn., Colin Wilson scored at 2:08 in the extra period, and the Predators beat San Jose to snap a two-game skid.
Pekka Rinne made 25 saves for his second shutout this season in a game in which Nashville set a franchise mark for offensive futility before Wilson scored. The goal ended the drought at just over 176 minutes in a three-game span.
FLYERS 4, JETS 3
In Winnipeg, Brayden Schenn, Kimmo Timonen and Tye McGinn scored to lead Philadelphia.
The Flyers (6-7-1), coming off a 5-2 loss in Toronto, won on the road for the second time this season. Jakub Voracek had two assists.